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INVERSE PROBLEMS IN GEOPHYSICS

ROEL SNIEDER AND JEANNOT TRAMPERTDept. of Geophysics Utrecht UniversityP.O. Box 80.021 3508 TA UtrechtThe Netherlands email [email protected]

1. Introduction An important aspect of the physical sciences is to make inferences aboutphysical parameters from data. In general, the laws of physics provide the

means for computing the data values given a model. This is called the"forward problem", see figure 1. In the inverse problem, the aim is to reconstruct the model from a set of measurements. In the ideal case, anexact theory exists that prescribes how the data should be transformed in order to reproduce the model. For some selected examples such a theoryexists assuming that the required infinite and noise-free data sets would be available. A quantum mechanical potential in one spatial dimension canbe reconstructed when the reflection coefficient is known for all energies [Marchenko, 1955; Burridge, 1980]. This technique can be generalized forthe reconstruction of a quantum mechanical potential in three dimensions [Newton, 1989], but in that case a redundant data set is required for rea-sons that are not well understood. The mass-density in a one-dimensional string can be constructed from the measurements of all eigenfrequencies ofthat string [Borg, 1946], but due to the symmetry of this problem only the even part of the mass-density can be determined. If the seismic velocityin the earth depends only on depth, the velocity can be constructed exactly from the measurement of the arrival time as a function of distance

0Present address of R. Snieder: Dept. of Geophysics, Colorado School of Mines, Golden CO 80401, USA

0Reprinted from "Wavefield Inversion", Ed. A. Wirgin, Springer Verlag, New York,

p. 119-190, 1999.

120

Data dModel m Forward problem

Inverse problem Figure 1. The traditional definition of the forward and inverse problems.

of seismic waves using an Abel transform [Herglotz, 1907; Wiechert, 1907].Mathematically this problem is identical to the construction of a spherically symmetric quantum mechanical potential in three dimensions [Kelleret al., 1956]. However, the construction method of Herglotz-Wiechert only gives an unique result when the velocity increases monotonically with depth[Gerver and Markushevitch, 1966]. This situation is similar in quantum mechanics where a radially symmetric potential can only be constructeduniquely when the potential does not have local minima [Sabatier, 1973].

Despite the mathematical elegance of the exact nonlinear inversionschemes, they are of limited applicability. There are a number of reasons for this. First, the exact inversion techniques are usually only applicablefor idealistic situations that may not hold in practice. For example, the Herglotz-Wiechert inversion presupposes that the velocity in the earth de-pends only on depth and that the velocity increases monotonically with depth. Seismic tomography has shown that both requirements are not metin the earth's mantle [Nolet et al., 1994]. Second, the exact inversion techniques often are very unstable. The presence of this instability in the solu-tion of the Marchenko equation has been shown explicitly by Dorren et al. [1994]. However, the third reason is the most fundamental. In many inverseproblems the model that one aims to determine is a continuous function of the space variables. This means that the model has infinitely many degreesof freedom. However, in a realistic experiment the amount of data that can be used for the determination of the model is usually finite. A simplecount of variables shows that the data cannot carry sufficient information to determine the model uniquely. In the context of linear inverse problemsthis point has been raised by Backus and Gilbert [1967, 1968] and more recently by Parker [1994]. This issue is equally relevant for nonlinear inverseproblems.

The fact that in realistic experiments a finite amount of data is availableto reconstruct a model with infinitely many degrees of freedom necessarily

121 means that the inverse problem is not unique in the sense that there aremany models that explain the data equally well. The model obtained from the inversion of the data is therefore not necessarily equal to the true modelthat one seeks. This implies that the view of inverse problems as shown in figure 1 is too simplistic. For realistic problems, inversion really consistsof two steps. Let the true model be denoted by

m and the data by d.From the data d one reconstructs an estimated model ~m, this is called theestimation problem, see figure 2. Apart from estimating a model ~

m thatis consistent with the data, one also needs to investigate what relation the

estimated model ~m bears to the true model m. In the appraisal problem onedetermines what properties of the true model are recovered by the estimated model and what errors are attached to it. The essence of this discussion isthat inversion = estimation + appraisal. It does not make much sense to make a physical interpretation of a model without acknowledging the factof errors and limited resolution in the model [Trampert, 1998].

Data d True model m Forward problem

Estimation problem Appraisal problem